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Democrats challenged Mr. Boehner to call up a “clean” continuing resolution to reopen and fund the government through Nov. 15.

Mr. Boehner, though, said flatly that a “clean” CR wouldn’t pass the House — though he hasn’t put that bill on the floor to prove it.

Democrats have countered that more than 17 House Republicans have said they would vote for a “clean” CR, which means that if all House Democrats also voted for it, there would in fact be a majority.

Sen. Charles E. Schumer, New York Democrat, said if Mr. Boehner is so certain of his math, he should hold a vote and prove it.

“Put it on the floor and let’s see if you’re right,” he said.

Meeting over the weekend, the House passed a bill to guarantee that all federal employees get paid for the shutdown — even those who have been on furlough. But they, like most employees still on the job, will have to wait until funding is returned to collect paychecks.

The bill cleared on a 407-0 vote, showing a striking sense of bipartisanship amid the contentious debate.

Mr. Obama has issued a statement supporting the legislation, though it’s not clear whether Mr. Reid will bring the bill to the Senate floor.

He again said he will not fund the government piece by piece and insisted that House Republicans vote to reopen all of government or else none of it will be reopened.

“A piecemeal approach of funding the government bit by bit is simply not the answer and saying, ‘We are going to pay you when we reopen this place’ — that is not the answer,” he said on the floor of the Senate. “No matter how many bites Republicans take at the apple, there is only one bill that ensures that every federal priority is met — the Senate bill that would fully reopen the federal government.”